This paper offers an overview of various alternative formulations for Analysis, the theory of Integral and Differential Calculus, and its diverging conceptions of the topological structure of the continuum. We pay particularly attention to Smooth Analysis, a proposal created by William Lawvere and Anders Kock based on Grothendieck’s work on a categorical algebraic geometry. The role of Heyting’s logic, common to all these alternatives is emphasized.

We present a solution to the problem of defining a counterpart in Algebraic Set Theory of the construction of internal sheaves in Topos Theory. Our approach is general in that we consider sheaves as determined by Lawvere-Tierney coverages, rather than by Grothendieck coverages, and assume only a weakening of the axioms for small maps originally introduced by Joyal and Moerdijk, thus subsuming the existing topos-theoretic results.

We suggest a new approach for addressing the problem of establishing an axiomatic foundation for large cardinals. An axiom asserting the existence of a large cardinal can naturally be viewed as a strong Axiom of Infinity. However, it has not been clear on the basis of our knowledge of ω itself, or of generally agreed upon intuitions about the true nature of the mathematical universe, what the right strengthening of the Axiom of Infinity is—which large cardinals ought to be derivable? (...) It was shown in the 1960s by Lawvere that the existence of an infinite set is equivalent to the existence of a certain kind of structure-preserving transformation from V to itself, not isomorphic to the identity. We use Lawvere's transformation, rather than ω, as a starting point for a reasonably natural sequence of strengthenings and refinements, leading to a proposed strong Axiom of Infinity. A first refinement was discussed in later work by Trnková—Blass, showing that if the preservation properties of Lawvere's tranformation are strengthened to the point of requiring it to be an exact functor , such a transformation is provably equivalent to the existence of a measurable cardinal. We propose to push the preservation properties as far as possible, short of inconsistency. The resulting transformation V→V is strong enough to account for virtually all large cardinals, but is at the same time a natural generalization of an assertion about transformations V→V known to be equivalent to the Axiom of Infinity. (shrink)

0. The ancient and honorable role of philosophy as a servant to the learning, development and use of scientific knowledge, though sadly underdeveloped since Grassmann, has been re-emerging from within the particular science of mathematics due to the latter's internal need; making this relationship more explicit (as well as further investigating the reasons for the decline) will, it is hoped, help to germinate the seeds of a brighter future for philosophy as well as help to guide the much wider learning (...) of mathematics and hence of all the sciences. 1. The unity of interacting opposites "space vs. quantity", with the accompanying "general vs. particular" and the resulting division of variable quantity into the interacting opposites "extensive vs. intensive", is susceptible, with the aid of categories, functors, and natural transformations, of a formulation which is on the one hand precise enough to admit proved theorems and considerable technical development and yet is on the other hand general enough to admit incorporation of almost any specialized hypothesis. Readers armed with the mathematical definitions of basic category theory should be able to translate the discussion in this section into symbols and diagrams for calculations. 2. The role of space as an arena for quantitative "becoming" underlies the qualitative transformation of a spatial category into a homotopy category, on which extensive and intensive quantities reappear as homology and cohomology. 3. The understanding of an object in a spatial category can be approached through definite Moore-Postnikov levels; each of these levels constitutes a mathematically precise "unity and identity of opposites", and their ensemble bears features strongly reminiscent of Hegel's Science of Logic. This resemblance suggests many mathematical and philosophical problems which now seem susceptible of exact solution. (shrink)

Foundations and Applications depend ultimately for their existence on each other. The main links between them are education and the axiomatic method. Those links can be strengthened with the help of a categorical method which was concentrated forty years ago by Cartier, Grothendieck, Isbell, Kan, and Yoneda. I extended that method to extract some essential features of the category of categories in 1965, and I apply it here in section 3 to sketch a similar foundation within the smooth categories which (...) provide the setting for the mathematics of change. The possibility that other methods may be needed to clarify a contradiction introduced by Cantor, now embedded in mathematical practice, is discussed in section 5. (shrink)

For 20th century mathematicians, the role of Cantor's sets has been that of the ideally featureless canvases on which all needed algebraic and geometrical structures can be painted. (Certain passages in Cantor's writings refer to this role.) Clearly, the resulting contradication, 'the points of such sets are distinc yet indistinguishable', should not lead to inconsistency. Indeed, the productive nature of this dialectic is made explicit by a method fruitful in other parts of mathematics (see 'Adjointness in Foundations', Dialectia 1969). This (...) role of Cantor's theory is compared with the role of Galois theory in algebraic geometry. (shrink)

Name der Zeitschrift: Zeitschrift für kritische Sozialtheorie und Philosophie Jahrgang: 5 Heft: 1 Seiten: 4-13.

ArgumentThis paper aims at contributing to the ongoing efforts to get a firmer grasp of the systematic significance of the entanglement of idealism and empiricism in Helmholtz's work. Contrary to existing analyses, however, the focal point of the present exposition is Helmholtz's attempt to articulate a psychological account of objectification. Helmholtz's motive, as well as his solution to the problem of the object are outlined, and interpreted against the background of his scientific practice on the one hand, and that of (...) empiricist and (transcendental) idealist analyses of experience on the other. The specifically psychological angle taken, not only prompts us to consider figures who have hitherto been treated as having only minor import for Helmholtz interpretation (most importantly J.S. Mill and J.G. Fichte), it furthermore sheds new light on some central tenets of the latter's psychological stance that have hitherto remained underappreciated. For one thing, this analysis reveals an explicit voluntarist tendency in Helmholtz's psychological theory. In conclusion, it is argued that the systematic significance of Helmholtz's empirico-transcendentalism with respect to questions of the mind is best understood as an attempt to found his empirical theory of perception in a second order, normative account of epistemic subjectivity. (shrink)

Explicit concepts and sufficiently precise definitions are the basis for further advance of a science beyond a given level. To move toward a situation where the whole population has access to the authentic results of science (italics mine) requires making explicit some general philosophical principles which can help to guide the learning, development, and use of mathematics, a science which clearly plays a pivotal role regarding the learning, development and use of all the sciences. Such philosophical principles have not come (...) from speculation but from studying and concentrating the development of actual mathematical subjects such as algebraic geometry, functional analysis, continuum mechanics, combinatorics, etc. The simplifications in the conceptual relations revealed by such philosophical/mathematical advances are of great value, not only in: (1) clarifying and guiding further research in the mathematical sciences, but also in (2) reforming pedagogy and popularization in ways that will not prove deceptive. (shrink)

Nineteenth-century scientist Hermann von Helmholtz’s peculiar wavering between empiricism and transcendentalism in his philosophy of science in general, and in his theory of perception in particular, is a much debated and well-documented topic in the history and philosophy of science. This contribution aims at providing a fresh angle on this classical issue, by considering Helmholtz’s account of differential consciousness against the background of a centuries-old philosophical debate between the empiricist tradition and the tradition of transcendental idealism. By placing Helmholtz’s psychology (...) against the background of a historical narrative stretching from Hume to Fichte, one can gain insight into the possible merits of his empirico-transcendentalism with regard to the problem of differentiation. More particularly, it is argued that Helmholtz’s psychology tilted towards transcendentalism when met with the classical theoretical problems of strict empiricism in dealing with the foundation of consciousness, most notably circularity and infinite regress. Without claiming that Helmholtz’s theorizing presented a self-conscious attempt to overcome the latter issues, his well-known wavering between perspectives in general, and his appropriation of the a priori in particular, might have served him well in avoiding the deadlocks of empiricism. As noted at the end, however, Helmholtz’s account produced complex philosophical problems of its own. (shrink)

Der Mensch ist fähig zu erkennen. Diese Fähigkeit setzt allerdings seine bio logische Integrität (Ganzheit, Unversehrtheit) voraus. Der Mensch kann außerdem erkennen, daß er erkennt. Erkennen (Kognition) als basale psychologische und somit biologische Funktion steuert seine Handhabung der Welt, und Wissen gibt seinen Handlungen Sicherheit. Objektives Wissen scheint möglich, und die Welt erscheint dadurch planvoll und vorhersagbar. Und doch ist Wissen als Erfahrung etwas Persönliches und Privates, das nicht übertragen werden kann. Das, was man für übertragbar hält, nämlich objektives Wissen, (...) muß immer durch den Hörer geschaffen werden: Der Hörer versteht nur dann, und objektives Wissen erscheint nur dann übertragbar, wenn der Hörer zu verstehen (vor)bereit(et) ist. Kognition als biologische Funktion besteht darin, daß sich die Antwort auf die Frage Was ist Kognition? aus dem Verständnis des Erkennens bzw. des Erkennenden aufgrund der Fähigkeit des letzteren zu erkennen ergeben muß. Eben darin besteht meme Arbeit. Epistemologie Der zentrale Anspruch der Wissenschaft ist Objektivität: sie [die Wissenschaft] ist bestrebt, mit Hilfe einer wohldefinierten Methodologie Aussagen über die Welt zu machen. Gerade in der Basis dieses Anspruches liegt jedoch ihre Schwäche: die apriorische Annahme, daß objektives Wissen eine Beschreibung dessen darstellt, was man weiß. Eine solche Annahme erfordert die Klärung der Fragen Worin be steht Erkennen bzw. Wissen? und Wie erkennen und wissen wir? Biologie a) Das größte Hindernis für das Verständnis der Organisation des Lebendigen liegt darin, daß es nicht möglich ist, sie durch eine Aufzählung ihrer Eigen schaften zu erklären. Sie muß vielmehr als Einheit verstanden werden. (shrink)

This review article situates the work Black Flame within a capita selecta of earlier publications on anarchism-syndicalism and radical thought. Schmidt and Van der Walt's contribution (2009) is a recent addition to political thought, theory and socio-economic practice within the broad stream of anarcho-syndicalism. Its treatment of anarchism and anarchist syndicalist groups in the workplace within an international context since the middle 1800s and the attempt to situate the debate in contemporary society are some notable features. The authors engage with (...) the past relevance of anarcho-syndicalism and implications for the future given current discontent with (global) politics. As an ongoing discourse on social philosophy, references are made to areas of agreement and differences between Anarchist and Marxist discourse. The latter versus one ‘grand narrative’ conjectures that liberal-capitalism on a global scale is bringing about an ‘End of History’ and that we have entered a post-industrial or even post-modern condition on the globe. However, theories-put-into-practice around the implosion of history and a ‘clash of civilisations’ result in deep inequality and the rise of absolute poverty. The work invites reflection and critical engagement. (shrink)

The paper calls for argumentation theory to learn from moral and political philosophy. Several thinkers in these fields help understand the occurrence of what we may call legitimate dissensus: enduring disagreement even between reasonable people arguing reasonably. It inevitably occurs over practical issues, e.g., issues of action rather than truth, because there will normally be legitimate arguments on both sides, and these will be incommensurable, i.e., they cannot be objectively weighed against each other. Accordingly, ‘inference,’ ‘validity,’ and ‘sufficiency’ are inapplicable (...) notions. Further distinctive features of pro and con arguments in practical argumentation are explored, and some corollaries are drawn regarding evaluative norms of legitimate dissensus. Examples from immigrationrelated public debates in Denmark are given. (shrink)

The aim of this paper is to present an in-depth inquiry into one of the most disregarded dimensions of Fichte’s philosophy, i.e., the systematic place of embodiment in his transcendental epistemology. Highlighting the necessarily embodied nature of the constitution of the notion of thinghood or being in Fichte’s philosophy could not only help pave the way for a more elegant understanding of the relation between idealism’s and phenomenology’s subject views, it likewise enables a more comprehensive insight into Fichte’s much debated (...) theory of subjectivity. Furthermore, Fichte’s transcendental account of the body provides one with a new vantage point from which to consider some classical interpretive issues, most notably those pertaining to Fichte’s peculiar methodology, his Ideal-Realism, and his take on the problem of explanatory circularity in trying to tackle the problem of the genesis of self-consciousness. (shrink)

Political debate is a distinctive domain in argumentation, characterized by these features: it is about proposals for action, not about propositions that may have a truth value; there may be good arguments on both sides; neither the proposal nor its rejection follows by necessity or inference; the pros and the cons generally cannot, being multidimensional and hence incommen- surable, be aggregated in an objective way; each audience member must subjectively compare and balance arguments on the two sides; eventual consensus between (...) the debaters is not a reasonable requirement. From all this follows a view of the rhetor’s special obligation in democratic, deliberative rhetoric on which it becomes crucial, in the interest of the audience, that political debaters acknowledge good arguments on the opposite side and explain why, on balance, they deem the arguments favoring their own side to be stronger. (shrink)

Douglas Walton has done extensive and valuable work on the concepts of presumption and practical reasoning. However, Walton’s attempt to model practical reasoning as presumptive is misguided. The notions of “inference” and of the burden of proof shifting back and forth between proponent and respondent are misleading and lead to counterintuitive consequences. Because the issue in practical reasoning is a proposal, not a proposition, there are, in the standard case, several perfectly good reasons on both sides simultaneously, which implies that (...) argument appraisal necessarily contains a subjective element—a fact argumentation theory needs to conceptualize. (shrink)

The aim of this paper is to present an in-depth inquiry into one of the most disregarded dimensions of Fichte’s philosophy, i.e., the systematic place of embodiment in his transcendental epistemology. Highlighting the necessarily embodied nature of the constitution of the notion of thinghood or being in Fichte’s philosophy could not only help pave the way for a more elegant understanding of the relation between idealism’s and phenomenology’s subject views, it likewise enables a more comprehensive insight into Fichte’s much debated (...) theory of subjectivity. Furthermore, Fichte’s transcendental account of the body provides one with a new vantage point from which to consider some classical interpretive issues, most notably those pertaining to Fichte’s peculiar methodology, his Ideal-Realism, and his take on the problem of explanatory circularity in trying to tackle the problem of the genesis of (bodily) self-consciousness. (shrink)

In this article an attempt is made to develop a concept: the teenwoordige-So, which has a two-fold nature. On the one hand it grows from the philosophical dimensions of the work of Breyten Breytenbach, and on the other, once it is identified it can be used as a method for reading his work. But in order to determine the “environment” of the concept three main elements are investigated in this article. The first is the notion of the rhizome, the second (...) element is identified in/as the Zen Buddhist concept of Tathata or suchness, and the third revolves around an investigation of Jean-Luc Nancy's treatment of the notion of presence in his work The birth to presence. It is argued that both these concepts portray something of a condition of continuous birth or becoming and that this “sense” of the world finds specific expression in the literary work of Breyten Breytenbach. It is also proposed that the expression Breytenbach gives to the condition of presence and suchness opens a nomadic territory, or nomadic understanding of the world which has a political meaning. (shrink)

bespreking van: Marc Slors, Dat had je gedacht! Brein, bewustzijn en vrije wil in filosofisch perspectief, Amsterdam, Boom, 2012, 201 blz, ISBN 978 94 6105 7785.

If there is a specifically rhetorical approach to argumentation, I believe it is one that studies argumentation that is specifically rhetorical. So if we want to ask, “What is the rhetorical approach to argumentation?” we should first ask, “What is rhetorical argumentation?” It is worthwhile focusing on this question because various misleading definitions of rhetorical argumentation have been in circulation for almost as long as rhetoric has existed. Some misleading definitions see the defining property of rhetorical argumentation in the arguer’s (...) aim. And that aim, which is often assumed to override all the arguer’s other considerations, is strategic: to persuade his hearer(s) by any available means and .. (shrink)

Literature on corporate social responsibility has tended to treat economic benefits to the firm as unintentional spillovers that result from laudable CSR behavior. Empirical studies of the relationship between CSR and corporate financial performance have reported mixed findings. This article shifts the conceptual and empirical focus to investigate the conditions under which intentional profit-seeking through corporate social action projects can create economic value for the firm. The article uses resource-dependency theory and the resource-based view to define the firm’s external and (...) internal environments respectively. From that perspective, the article looks at how corporate social action creates economic value through strategic social planning and strategic social positioning. A survey instrument was developed and applied to 110 large Spanish firms. In that sample, munificence and continuous innovation positively affect social positioning, while nongovernmental organization salience and social responsibility orientation positively affect social planning. Both social positioning and social planning in turn contribute to corporate ability to create value. The article concludes with a discussion of the research and managerial implications of these findings. (shrink)

This introduction presents the set of six articles, written by rhetorical scholars, which constitute the bulk of the present special issue of Argumentation. In the introduction, the issue editor seeks to identify defining features of a rhetorical approach to argumentation. Taking this approach means dealing with argumentation in the “realm of rhetoric”, which comprises argumentation where deductive “demonstration” is not available. This has several corollaries, including the condition of uncertainty and the necessity of securing adherence from an audience. The articles (...) in the issue explore these and other characteristics of argumentation in the realm of rhetoric. (shrink)

Rapid individual cognitive phenotyping holds the potential to revolutionize domains as wide‐ranging as personalized learning, employment practices, and precision psychiatry. Going beyond limitations imposed by traditional lab‐based experiments, new efforts have been underway toward greater ecological validity and participant diversity to capture the full range of individual differences in cognitive abilities and behaviors across the general population. Building on this, we developed Skill Lab, a novel game‐based tool that simultaneously assesses a broad suite of cognitive abilities while providing an engaging (...) narrative. Skill Lab consists of six mini‐games as well as 14 established cognitive ability tasks. Using a popular citizen science platform (N = 10,725), we conducted a comprehensive validation in the wild of a game‐based cognitive assessment suite. Based on the game and validation task data, we constructed reliable models to simultaneously predict eight cognitive abilities based on the users’ in‐game behavior. Follow‐up validation tests revealed that the models can discriminate nuances contained within each separate cognitive ability as well as capture a shared main factor of generalized cognitive ability. Our game‐based measures are five times faster to complete than the equivalent task‐based measures and replicate previous findings on the decline of certain cognitive abilities with age in our large cross‐sectional population sample (N = 6369). Taken together, our results demonstrate the feasibility of rapid in‐the‐wild systematic assessment of cognitive abilities as a promising first step toward population‐scale benchmarking and individualized mental health diagnostics. (shrink)

Do there exist real numbers d with d2 = 0? The question is formulated provocatively, to stress a formalist view about existence: existence is consistency, or better, coherence.Also, the provocation is meant to challenge the monopoly which the number system, invented by Dedekind et al., is claiming for itself as THE model of the geometric line. The Dedekind approach may be termed “arithmetization of geometry”.We know that one may construct a number system out of synthetic geometry, as Euclid and followers (...) did : “geometrization of arithmetic”..Starting from the geometric side, nilpotent elements are somewhat reasonable, although Euclid excluded them. The sophist Protagoras presented a picture of a circle and a tangent line; the apparent little line segment D which tangent and circle have in common, are, by Pythagoras' Theorem, precisely the points, whose abscissae d have d2 = 0. Protagoras wanted to use this argument for destructive reasons: to refute the science of geometry.A couple of millenia later, the Danish geometer Hjelmslev revived the Protagoras picture. His aim was more positive: he wanted to describe Nature as it was. According to him, the Real Line, the Line of Sensual Reality, had many nilpotent infinitesimals, which we can see with our naked eyes. (shrink)